Methods and apparatus for splitting, imaging, and measuring wavefronts in interferometry

ABSTRACT

Apparatus for splitting, imaging, and measuring wavefronts with a reference wavefront and an object wavefront. A wavefront-combining element receives and combines into a combined wavefront an object wavefront from an object and a reference wavefront. A wavefront-splitting element splits the combined wavefront into a plurality of sub-wavefronts in such a way that each of the sub-wavefronts is substantially contiguous with at least one other sub-wavefront. The wavefront-splitting element may shift the relative phase between the reference wavefront and the object wavefront of the sub-wavefronts to yield a respective plurality of phase-shifted sub-wavefronts. The wavefront-splitting element may then interfering the reference and object wavefronts of the phase-shifted sub-wavefronts to yield a respective plurality of phase-shifted interferograms. An imaging element receives and images the phase-shifted interferograms. A computer connected to the imaging element measures various parameters of the objects based on the phase-shifted interferograms. Examples of measurements include flow parameters such as the concentrations of selected gaseous species, temperature distributions, particle and droplet distributions, density, and so on. In addition to flow parameters, the displacement (e.g., the vibration) and the profile of an object may be measured.

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] The present application is a divisional application of U.S.patent application Ser. No. 09/413,829 filed Oct. 6, 1999.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

[0002] This invention was made with Government support under ContractNo. DMI-9531391 awarded by the National Science Foundation. TheGovernment has certain rights in this invention.

BACKGROUND OF THE INVENTION

[0003] 1. Field of the Invention

[0004] The present invention relates to interferometry. Moreparticularly, the present invention relates to methods and apparatus forimaging wavefronts. The methods and apparatus of the present inventionmay be implemented in measuring systems that measure various parametersof test objects by simultaneously generating a plurality ofphase-shifted interferograms.

[0005] 2. Description of the Related Art

[0006] Phase-shift interferometry is an established method for measuringa variety of physical parameters ranging from the density of gasses tothe displacement of solid objects. Interferometric wavefront sensors canemploy phase-shift interferometers to measure the spatial distributionof relative phase across an area and, thus, to measure a physicalparameter across a two-dimensional region. An interferometric wavefrontsensor employing phase-shift interferometry typically consists of aspatially coherent light source that is split into two wavefronts, areference wavefront and an object wavefront, which are later recombinedafter traveling different optical paths of different lengths. Therelative phase difference between the two wavefronts is manifested as atwo-dimensional intensity pattern known as an interferogram. Phase-shiftinterferometers typically have an element in the path of the referencewavefront which introduces three or more known phase steps or shifts. Bydetecting the intensity pattern with a detector at each of the phaseshifts, the phase distribution of the object wavefront can bequantitatively calculated independent of any attenuation in either ofthe reference or object wavefronts. Both continuous phase gradients anddiscontinuous phase gradients (speckle waves) can be measured using thistechnique.

[0007] Temporal phase shifting using methods such as piezo-electricdriven mirrors have been widely used to obtain high-quality measurementsunder otherwise static conditions. The measurement of transient orhigh-speed events requires either ultra high-speed temporal phaseshifting (i.e., much faster than the event timescales), which is limiteddue to detector speed, or spatial phase shifting that can acquireessentially instantaneous measurements.

[0008] Several methods of spatial phase shifting have been disclosed inthe prior art. In 1983 Smythe and Moore described a spatialphase-shifting method in which a series of conventional beam splittersand polarization optics are used to produce three or four phase-shiftedimages onto as many cameras for simultaneous detection. A number ofUnited States patents, such as U.S. Pat. Nos. 4,575,248; 5,589,938;5,663,793; 5,777,741; and 5,883,717, disclose variations of the Smytheand Moore method where multiple cameras are used to detect multipleinterferograms. One of the disadvantages of these methods is thatmultiple cameras are required and complicated optical arrangements areneed to produce the phase-shifted images, resulting in expensive complexsystems.

[0009] Other methods of spatial phase shifting include the use ofgratings to introduce a relative phase step between the incident anddiffracted beams, an example of which is disclosed in U.S. Pat. No.4,624,569. However, one of the disadvantages of these grating methods isthat careful adjustment of the position of the grating is required tocontrol the phase step between the beams.

[0010] Spatial phase shifting has also been accomplished by using atilted reference wave to induce a spatial carrier frequency to thepattern, an example of which is disclosed in U.S. Pat. No. 5,155,363.This method requires the phase of the object field to vary slowly withrespect to the detector pixels; therefore, using this method withspeckle fields requires high magnification.

[0011] Yet another method for measuring the relative phase between twobeams is disclosed in U.S. Pat. No. 5,392,116, in which a linear gratingand four detector elements are used. This method has a number ofdrawbacks, including the inability to measure of wavefronts (i.e., thespatial phase distribution across the profile of a beam) and to formcontiguous images on a single pixilated detector such as a standardcharge coupled device (CCD).

[0012] Finally, it is noted that wavefront sensing can be accomplishedby non-interferometric means, such as with Shack-Hartmann sensors whichmeasure the spatially dependent angle of propagation across a wavefront.These types of sensors are disadvantageous in that they typically havemuch less sensitivity and spatial resolution than interferometricwavefront sensors and are not capable of performing relative phasemeasurements such as two-wavelength interferometry.

BRIEF SUMMARY OF THE INVENTION

[0013] It is one object of the present invention to provide aninterferometric wavefront sensor that incorporates spatial phaseshifting but avoids the complexity of multi-camera systems by using asingle two-dimensional pixilated detector, such as a standard chargecoupled device (CCD) camera.

[0014] It is another object of the present invention to provide methodsand apparatus for performing two-wavelength interferometry that utilizea compact spatial phase-shifting device to acquire data at high speedsand provide improved tolerance to vibration.

[0015] It is yet another object of the invention to provide methods andapparatus for dividing an incoming wavefront into four sub-wavefrontsthat are imaged substantially contiguous to maximize the coverage of apixilated area detector, while minimizing the number of necessaryoptical components to provide a compact system.

[0016] It is still another object of the invention to provide methodsand apparatus for introducing a phase shift between orthogonallypolarized reference and object wavefronts that is uniform across eachsub-wavefront and not sensitive to the positioning of a diffractiveoptical element.

[0017] According to one aspect of the invention, apparatus for splittinga wavefront and producing four substantially contiguous images of thewavefront consists of an input plane, a first lens element, adiffractive optical element, a second lens element, and an output plane.The lens elements are placed in a telescopic arrangement (separated bythe sum of their focal lengths) and the diffractive optical element isplaced at or near the mutual focal points. The diffractive opticalelement produces four output wavefronts (or beams) from a single inputwavefront. In a preferred embodiment the diffractive element producesfour diffracted orders of equal intensity and symmetric to the incidentaxis so that it can be characterized by a single divergence angle a anda radial angular spacing of β. The diffractive optic is constructed tosuppress the zero order component to the greatest extent possible.Alternatively, the diffractive optical element may produce threediffracted orders each of equal intensity with the transmitted zeroorder beam. The diffractive optic may include a wedged substrate toprovide a uniform angular tilt to all four beams so they propagatesymmetrically to the axis of the incident beam. Again, the compounddiffractive optical element is characterized by a single divergenceangle α and a radial angular spacing β. Any higher-order diffractedcomponents from the diffractive optic should be at least twice theangular divergence. The focal length of the second lens may be selectedto be equal to the detector size divided by two times the tangent of thediffractive optic's divergence angle. The front lens may be chosen toproduce an overall system magnification equivalent to the originalwavefront dimension divided by half the detector size.

[0018] According to another aspect of the invention, apparatus forintroducing a uniform phase-shift between orthogonally polarizedreference and object wavefronts includes a polarization mask elementmade of discrete sections. Each section includes a phase retardationplate or a blank and a linear polarizer. The relative angularorientation of the phase retardation plate and linear polarizer isselected to be different for each discrete section. In one exemplaryembodiment, the mask element includes four quadrants each providing aphase shift of π/2 relative to the clockwise adjacent quadrant.

[0019] According to still another aspect of the present invention, asystem for providing an improved wavefront sensor includes a wavefrontsplitting element, a polarization mask element, a pixilated detectorelement, a polarization interferometer, and a computer. The phase of anobject beam can be measured with a single frame of data acquired fromthe pixilated detector.

[0020] Yet another aspect of the invention provides a two-wavelengthinterferometer including a wavefront sensor with a tunable laser ormultiple laser sources. Multiple wavefronts are measured at each ofseveral wavelengths with the relative phase values subtracted todetermine the contour of an object.

[0021] Other objects, features, and advantages of the present inventionwill become apparent to those skilled in the art from a consideration ofthe following detailed description taken in conjunction with theaccompanying drawings.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0022]FIG. 1 is a schematic view of measurement apparatus configured inaccordance with the present invention, particularly illustrating themeasurement apparatus with the use of functional blocks;

[0023]FIG. 2 is a schematic perspective view of an exemplary embodimentof apparatus for generating multiple phase-shifted images in accordancewith the present invention;

[0024]FIG. 3 is a schematic perspective view of an exemplaryphase-retardant plate according to the invention, particularlyillustrating a phase-retardant plate for shifting the phase of fourwavefronts;

[0025]FIG. 4 is a plan view of the phase-retardant plate shown in FIG.3;

[0026]FIG. 5 is a schematic view of an exemplary embodiment ofmeasurement apparatus of the invention, particularly illustratingtransmit and image portions thereof;

[0027]FIG. 6 is a schematic view of an exemplary embodiment of an imageportion of the measurement apparatus of the invention;

[0028]FIG. 7 is a schematic view of an active surface of a detectorarray of an image portion of the present invention, particularlyillustrating an exemplary plurality of sub-wavefronts coaxial along anoptical axis of the image portion;

[0029]FIG. 8 is a schematic view of another exemplary embodiment of animaging portion of the present invention, particularly illustrating theinclusion of a polarizer and a mask;

[0030]FIG. 9 is a schematic view illustrating an exemplary imagingportion of the invention;

[0031]FIG. 10 is a schematic view of another exemplary embodiment ofmeasurement apparatus of the invention, particularly illustratingapparatus for performing profilometry;

[0032]FIG. 11 is a schematic view of the measurement apparatus of FIG.6, particularly illustrating an exemplary commercial embodiment of theprofilometer of the invention;

[0033]FIG. 12 is a schematic view of a yet another exemplary embodimentof measure apparatus of the invention, particularly illustratingapparatus for measuring displacement;

[0034]FIG. 13 is a schematic view of still another exemplary embodimentof the measurement apparatus of the invention, particularly illustratingapparatus for performing wavefront sensing; and

[0035]FIG. 14 is a schematic view of a graphical user interfaceillustrating interferometric data according to the present invention.

DETAILED DESCRIPTION OF THE INVENTION

[0036] The present invention provides apparatus and methodology formeasuring various parameters of test objects by simultaneouslygenerating multiple phase-shifted images. More particularly, theapparatus and methodology of the present invention enable multiplephase-shifted images (or interferograms) to be obtained with a singleimaging device and by a single pulse of a laser and at very high rates.In doing so, the present invention splits, images, and measures awavefront made up of a reference and an object wavefront from an objectunder test.

[0037] The apparatus of the present invention may be configured tomeasure—in situ and in real time—flow parameters in a multiphaseenvironment. Examples of such flow parameters include the concentrationsof selected gaseous species, temperature distributions, particle anddroplet distributions, density, and so on. In addition to flowparameters, the apparatus of the present invention may be configured tomeasure the displacement (e.g., the vibration) of an object. Moreover,the apparatus of the invention may be configured to perform profilometryof an object, that is, to measure the absolute three-dimensionalprofiles of solid objects. These and other utilizations and embodimentsof the technology of the present invention are discussed in detailherein.

[0038] Turning to the drawings, a measurement system 50 exemplifying theprinciples of the present invention is illustrated in FIG. 1. Exemplarymeasurement system 50 generally includes a transmit portion 52 and animage portion 54. The transmit portion 52 transmits a referencewavefront 56 to the image portion 54 and an object wavefront 58 to anobject 60 under measurement. The reference and object wavefronts 56 and58 are preferably generated by a spatially coherent light source such asa laser. The object wavefront 58 is received by the image portion 54after acting upon the object 60, for example, by reflection or bytransmission. Data obtained by the image portion 54 from the object 60may be provided to a computer 62 for processing. The transmit portion 52and the image portion 54 may be oriented with respect to the object 60according to a plurality of measurement configurations, which arediscussed in detail below.

[0039] With continued reference to FIG. 1, exemplary image portiongenerally includes a wavefront-combining element 64 for receiving thereference wavefront 56 and the object wavefront 58 and for combining thewavefronts into a combined wavefront 66. The reference and objectwavefronts 56 and 58 are combined to be superimposed and orthogonallypolarized, which is discussed below. A wavefront-splitting element 68receives the combined wavefront 66 and splits the wavefront into aplurality of sub-wavefronts 70. A phase-shifting interference element 72receives the sub-wavefronts 70 and is configured to shift the relativephase between the reference and object wavefronts 56 and 58 and tointerfere the reference and object wavefronts 56 and 58 by polarization,for each of the sub-wavefronts 70, to yield a plurality of phase-shiftedinterferograms 74. A sensing element 76 receives the phase-shiftedinterferograms 74 from the phase-shifting interference element 72substantially simultaneously. The sensing element 76 provides data 78indicative of the interferograms 74 to the computer 62 for processing.

[0040] According to the present invention, the phase-shiftinginterference element 72 shifts the relative phase between the referenceand object wavefronts 56 and 58 for each of the sub-wavefronts 70discretely by a factor of a predetermined amount p. The predeterminedamount p may be determined by a number N of sub-wavefronts 70 in theplurality of sub-wavefronts generated by the wavefront-splitting element68 from the combined wavefront 66. For example, the predetermined amountp may be determined as the quotient of 360 degrees and the number N ofsub-wavefronts 70, or:

p−360°÷N.  (1)

[0041] Accordingly, the discrete phase shift Δφ of each of the pluralityof sub-wavefronts 70 may be determined as:

Δφ_(i)=(i−1)×p,  (2)

[0042] where i=1 to N. For example, if the wavefront-splitting element68 provides four sub-wavefronts 70, then the discrete phase shifts Δφ ofthe four wavefronts are 0°, 90°, 180°, and 270°. According to thisembodiment, there is a 90° phase shift between each of theinterferograms 74.

[0043] An exemplary embodiment of the combination of thewavefront-splitting element 68, the phase-shifting interference element72, and the sensing element 76 is illustrated in FIG. 2. As shown, thecombined wavefront 66 includes the reference wavefront 56 from thetransmit portion 52 and the object wavefront 58 from the object 60. Thewavefront-combining element 64 is configured so that the referencewavefront 56 and the object wavefront 58 are orthogonally polarized,which is indicated in FIG. 2 by the scientific convention of an arrowand a dot. Exemplary wavefront-splitting element 68 is preferably atwo-dimensional diffractive optical element (DOE) such as a holographicoptical element (HOE) 80. According to a preferred embodiment of theinvention, exemplary DOE 80 splits the combined wavefront 66 into foursub-wavefronts 70 a, 70 b, 70 c, 70 d. Each of the sub-wavefronts 70a-70 d follows a spatially discrete path.

[0044] With continued reference to FIG. 2, exemplary phase-shiftinginterference element 72 includes a plurality of sections 82, the numberof which preferably equals the number N of sub-wavefronts 70 provided bythe wavefront-splitting element 68. According to the preferredembodiment shown, exemplary phase-shifting interference element 72includes four sections 82 a, 82 b, 82 c, 82 d. The phase-shiftinginterference element 72 is disposed with respect to thewavefront-splitting element 68 so that the plurality of sub-wavefronts70 are respectively incident on the plurality of sections 82; that is,each section 82 receives one of the sub-wavefronts 70. As discussedabove, each of the sections 82 shifts the relative phase between thereference and object wavefronts 56 and 58 and interferes the twowavefronts 56 and 58 for each of the sub-wavefronts 70 incident thereonby a discrete phase shift Δφ. Each of the sections 82 a, 82 b, 82 c, . .. 82N of the phase-shifting interference element 72 accordingly providesa respective phase-shifted interferograms 74 a, 74 b, 74 c, . . . , 74N.The phase of each phase-shifted interferogram 74 is out of phase withthe phase of the other phase-shifted interferograms 74 by a factor ofthe predetermined amount p of phase shift, which is discussed furtherbelow.

[0045] Continuing to reference FIG. 2, exemplary sensing element 76 ispreferably an imaging sensor or a detector array 84. The detector array84 may be a video-imaging sensor such as a charged coupled device (CCD)camera. According to the present invention, the detector array 84preferably has an active surface 86. The active surface 86 may bedefined by a pixel array. The detector array 84 may be made from aplurality of individual detector arrays configured to function as asingle active sensing element. For example, the active surface 86 may bedefined by more than one CCDs collectively functioning as a singlearray. For the purposes of this description, the active surface 86 has asurface area S.

[0046] The detector array 84 is disposed with respect to thephase-shifting interference element 72 so that the plurality ofphase-shifted interferograms 74 are substantially simultaneouslyincident on the active surface 86, thereby imaging on the active surface86 a respective plurality of phase-shifted interferograms. Based on theimaged interferograms, the spatially resolved phase of each of thephase-shifted interferograms 74 can be measured instantaneously. Inaddition, the detector array 84 is disposed with respect to thephase-shifting interference element 72 so as to maximize the area of theactive surface 86, which is discussed in more detail below.

[0047] With additional reference to FIG. 3, an exemplary embodiment ofthe phase-shifting interference element 72 includes a plurality ofplates 88. For the preferred four-component embodiment described above,exemplary phase-shifting interference element 72 includes a first plate88 a and a second plate 88 b. For purposes of clarity and illustration,the plates 88 are shown in a spaced relationship; however, according toexemplary embodiments of the invention, the plates 88 are substantiallyplanar, disposed in a parallel relationship, and abut each other. Thefirst plate 88 a includes a quarter-wave plate 90 and a blank plate 92.As known in the art, a quarter waveplate shifts the relative phase oftwo orthogonally polarized incident wavefronts by 90°, and a blank plateshifts the relative phase of two orthogonally polarized incidentwavefronts by 0° (i.e., there is no relative phase shift). The plates 90and 92 are preferably coplanar and divide the first plate 88 a intorespective halves.

[0048] The second plate 88 b of exemplary phase-shifting interferenceelement 72 includes a pair of polarizing plates 94 a and 94 b that areconfigured to polarize an incident wavefront linearly so that electricfield vectors of the transmitted wavefront are perpendicular with eachother. Specific to the illustrated embodiment, one of the polarizingplates, e.g., plate 94 a, is configured to polarize light at +45° withrespect to the vertical axis (as shown by arrow A in FIG. 3), therebyinterfering the in-phase components of the reference and objectwavefronts 56 and 58. The other polarizing plate, e.g., plate 94 b, isconfigured to polarize light at −45° with respect to the vertical axis(as shown by arrow B in FIG. 3), thereby interfering the out-of-phasecomponents of the reference and object wavefronts 56 and 58. Thepolarizing plates 94 a and 94 b are preferably coplanar and divide thesecond plate 88 b into respective halves.

[0049] With continued reference to FIG. 3 and additional reference toFIG. 4, the first and second plates 88 a and 88 b are configured so thatthe respective halves thereof are perpendicular with each other, thusforming a phase-retardation mask or plate 96. In the four-componentembodiment shown, the phase-retardation plate 96 includes four sections82, each of which defines a quadrant. Section 82 a, or quadrant Q₀, isdefined by the blank plate 92 and polarizing plate 94 a, thusinterfering the in-phase (i.e., 0°) component between the incidentreference and object wavefronts 56 and 58. Section 82 b, or quadrant Q₁,is defined by the quarter-wave plate 90 and polarizing plate 94 a, thusinterfering the in-phase quadrature (i.e., 90°) component between theincident reference and object wavefronts 56 and 58. Section 82 c, orquadrant Q₂, is defined by the blank plate 92 and polarizing plate 94 b,thus interfering the out-of-phase (i.e., 180°) component between theincident reference and object wavefronts 56 and 58. And section 82 d, orquadrant Q₃, is defined by the quarter-wave plate 90 and polarizingplate 94 b, thus interfering the out-of-phase quadrature (i.e., 270°)component between the incident reference and object wavefronts 56 and58.

[0050] The operation of the phase-shifting interference element 72 maybe described with respect to the reference and object wavefronts 56 and58 which, as mentioned above, are orthogonally polarized. The electricfield vectors for each of the wavefronts 56 and 58 may be written as:

{right arrow over (E)} _(r) =Re ^(i(kz−ωt)) ŝ  (3a)

{right arrow over (E)} _(s) =Se ^(i(kz−wt+Δφ)) {circumflex over(p)}  (3b)

[0051] where:

[0052] R and S are the amplitudes of each wavefront 56 and 58,respectively;

[0053] is the optical frequency;

[0054] t is time;

[0055] k is the wavevector=2π/λ;

[0056] p and s are orthogonal unit polarization vectors; and

[0057] ΔΦ is the phase difference between the wavefronts 56 and 58.

[0058] The intensity (I) of each of the phase-shifted interferograms 74incident on the active surface 86 of the detector array 84 is given by:$\begin{matrix}{I_{0} = {\frac{1}{2}\left( {I_{r} + I_{s} + {2\sqrt{I_{r}I_{s}}{\cos \left( {\Delta \quad \varphi} \right)}}} \right)}} & \text{(4a)} \\{I_{1} = {\frac{1}{2}\left( {I_{r} + I_{s} + {2\sqrt{I_{r}I_{s}}{\cos \left( {{\Delta \quad \varphi} + \frac{\pi}{2}} \right)}}} \right)}} & \text{(4b)} \\{I_{2} = {\frac{1}{2}\left( {I_{r} + I_{s} + {2\sqrt{I_{r}I_{s}}{\cos \left( {{\Delta\varphi} + \pi} \right)}}} \right)}} & \text{(4c)} \\{I_{3} = {\frac{1}{2}\left( {I_{r} + I_{ss} + {2\sqrt{I_{r}I_{s}}{\cos \left( {{\Delta \quad \varphi} + \frac{3\pi}{2}} \right)}}} \right)}} & \text{(4d)}\end{matrix}$

[0059] where I_(r) and I_(s) are the intensities of the reference andobject wavefronts 56 and 58, respectively (which intensities areproportional to R² and S²). This set of phase-shifted intensities I₀,I₁, I₂, and I₃ may be analyzed numerically using a number of algorithmsto solve explicitly for the phase difference between the reference andobject wavefronts 56 and 58, which is discussed in detail below.

[0060] As it is preferable to maximize the imaging area of the detectorarray 84 (i.e., to maximize the portion of the surface area S of theactive surface 86 that is illuminated by the interferograms 74), thephase-retardation plate 96 is preferably disposed adjacent to orsubstantially at the active surface 86 of the detector array 84, whichis discussed in more detail below. By detecting the plurality ofphase-shifted interferograms 74 instantaneously with an imaging sensorexemplified by the detector array 84, the image portion 54 of theinvention enables the measuring system 50 to instantaneously measure theentire test object 60. In addition, the instantaneous detection of thephase-shifted interferograms 74 eliminates the need to scan individualbeams spatially through or across the surface of the object 60.

[0061] As mentioned above, exemplary measurement system 50 of thepresent invention may be configured in a plurality of preferredembodiments each designed to carry out a particular type of real-timemeasurement, including a profilometer, a displacement sensor, and awavefront sensor. In other words, exemplary embodiments of the measuringsystem 50 include a common transmit portion 52 and a common imageportion 54 that can be physically oriented in a plurality ofconfigurations with a plurality of optical and imaging components toundertake a plurality of measurements, which is discussed in detailbelow.

[0062]FIG. 5 illustrates one such exemplary configuration of themeasurement system 50 of the invention which may be used to performreal-time interferometry for measuring transient events. The transmitportion 52 according to this embodiment includes a coherent light sourcesuch as a laser or laser diode 98. The laser 98 may include a half-waveplate 100 to provide a coherent light wavefront 102 which is split by apolarizing beam splitter (PBS) 104 into the reference wavefront 56 andthe object wavefront 58. The PBS 104 is configured to provideorthogonally polarized wavefronts as shown. The object wavefront 58 isexpanded by, for example, a combination of an expanding lens 106 and acollimating lens 108. Upon expansion, the object wavefront 58 istransmitted to the test object 60 where the object wavefront 58 isincident upon the surface or boundary thereof and either reflected fromor transmitted through the object 60.

[0063] Exemplary image portion 54 receives the object wavefront 58 fromthe object 60 and may include optics for collimating the received objectwavefront 58, such as a combination of a collecting lens 110 and acollimating lens 112. Collimating lens 112 is preferably spaced from thecollecting lens 100 by a distance equal to the sum of their respectivefocal lengths f₁and f₂. The object wavefront 58 is then superimposedwith the reference wavefront 56 at the wavefront-combining element 64which may be a polarizing beam splitter (PBS) 114 to yield the combinedwavefront 66. PBS 114 is preferably spaced from collimating lens 112 bya focal length f₂ of the collimating lens. The combined wavefront 66 maybe focused on the diffractive optical element 80 by means of a convexlens 116. In turn, the plurality of sub-wavefronts 70 may be focused onthe phase-retardation/interference plate 96 either directly or by meansof a collimating lens 118 as shown.

[0064] The placement of the various elements with respect to each otheris chosen to maximize the operability of the image portion 54. Forexample, PBS 114, the convex lens 116, and the diffractive opticalelement 80 are preferably respectively spaced apart by focal length f₃,which is the focal length of the convex lens 116. In addition, thediffractive optical element 80, the collimating lens 118, and thephase-retardation/interference plate 96 are preferably respectivelyspaced apart by a focal length f₄, which is the focal length of thecollimating lens 118. The placement of the diffractive optical element80 at the focus of collimating lens 118, which is defined as the inputfocal plane or the Fourier transform plane, optimizes the area of theactive surface 86 of the detector array 84 illuminated by the pluralityof phase-shifted interferograms 74.

[0065] Referencing FIG. 6, the optics of exemplary imaging portion 54are shown in more detail. The optical elements of the imaging portion 54are aligned along an optical axis O. As mentioned above, the diffractiveoptical element 80 splits the combined wavefront 66 into a plurality of(e.g., four) sub-wavefronts 70. Each of the sub-wavefronts 70 follows anoptical path defined by the distance each of the sub-wavefronts 70follows from the diffractive optical element 80 to the active surface 86of the detector array 84.

[0066] The diffractive optical element 80 and lenses 116 and 118 areconfigured so that each of the imaged sub-wavefronts 70 incident atdetector surface 86 are adjacent to or substantially contiguous with atleast one other sub-wavefront, which is shown in FIG. 7. For example, inthe exemplary embodiment shown, sub-wavefront 70 a is substantiallycontiguous with sub-wavefronts 70 b and 70 c, which is respectivelyindicated by reference alphas AB and AC; sub-wavefront 70 b issubstantially contiguous with sub-wavefronts 70 a and 70 d, which isrespectively indicated by reference alphas AB and BD; sub-wavefront 70 cis substantially contiguous with sub-wavefronts 70 a and 70 d, which isrespectively indicated by reference alphas AC and CD; and sub-wavefront70 d is substantially contiguous with sub-wavefronts 70 b and 70 c,which is respectively indicated by reference alphas BD and CD. Thissubstantially contiguous nature of the sub-wavefronts 70 is furtherenhanced in an embodiment in which the diffractive optical element 80splits the combined wavefront 66 into a plurality of sub-wavefrontshaving a substantially rectangular cross section as shown in FIG. 8.

[0067] The exemplary diffractive optical element 80 preferably splitsthe combined wavefront 66 in such a manner that the sub-wavefronts 70diverge from the optical axis O at substantially equal angles. In apreferred embodiment, the diffractive optical element 80 may producefour diffracted orders that have equal intensity and are symmetric tothe incident axis so that the diffracted orders may be characterized bya single divergence angle α and a radial angular displacement β. Thediffractive optical element 80 may be constructed to suppress the zeroorder component to the greatest extent possible.

[0068] In another exemplary embodiment, the diffractive optical element80 may produce three diffracted orders each of equal intensity with thetransmitted zero order beam. The diffractive optical element 80 mayinclude a wedged substrate to provide a uniform angular tilt to all fourbeams so that the beams propagate symmetrically to the axis of theincident beam. As mentioned above, the diffractive optical element 80 ispreferably characterized by a single divergence angle α and a radialangular displacement β.

[0069] Referring to FIG. 7, the radial angular displacement β producedby exemplary diffractive optical element 80 is determined by the aspectratio of the height h and the width w of the active surface 86 of thedetector array 84. The desired radial angular displacement β is givenby: $\begin{matrix}{\beta = {2\quad {\tan^{- 1}\left( \frac{h}{w} \right)}}} & (5)\end{matrix}$

[0070] where w and h are the width and the height of the active surface86 of detector array 84. For a detector with a unity aspect ratio (i.e.,square), the radial angular displacement β becomes 90 degrees and allfour images are radially symmetric.

[0071] Accordingly, each of the sub-wavefronts 70 follows an independentoptical path from the diffractive optical element 80 to the activesurface 86 that has a length substantially equal to each of the otheroptical paths. As such, the plurality of sub-wavefronts 70 reach theactive surface 86 substantially simultaneously. By configuring theimaging portion 54 so that the sub-wavefronts 70 have substantiallyequal optical path lengths, the imaging portion 54 is less susceptibleto errors that may introduced by vibration to the system.

[0072] With particular reference to FIG. 7, exemplary active surface 86of the detector array 84 may have a plurality of sections 119 forrespectively receiving the plurality of sub-wavefronts 70. Each of thesections 119 has a surface area on which the respective sub-wavefront 70is incident. According to the present invention, the portion orpercentage of the surface area of each section 119 on which asub-wavefront is incident is preferably maximized, thereby maximizingthe resolution of the detector array 84. For example, each of thesub-wavefronts 70 a-70 d is incident on at least half of the surfacearea of a respective section 119 a-119 d. More preferably, thepercentage is at least 75%. In the embodiment shown in FIG. 7 by thecircular cross-hatched regions, the incident percentage of eachsub-wavefront 70 may be determined by πr² divided by (h/2+w/2)². In theembodiment shown in FIG. 7 by the rectangular cross hatched region, theincident percentage of each sub-wavefront is substantially 100%.

[0073] Further referencing FIG. 6 and with addition reference to FIG. 8,an aperture 121 may be provided at an input focal plane of the convexlens 116 (i.e., at a focal length f₃), with the diffractive opticalelement 80 positioned at the output focal plane of the convex lens 116.Alternatively, as shown in FIG. 9, a pair of apertures 121 a and 121 bmay be positioned upstream of PBS 114 through which the reference andobject wavefronts 56 and 58 respectively travel. According to apreferred embodiment of the invention, the aperture(s) 112 may berectangular with an aspect ratio substantially the same as the activesurface 86 of the detector array 84. The presence of the aperture(s) 121reduces the amount of ambient noise received in the image portion 54 andreduces crosstalk between the imaged sub-wavefronts.

[0074] An example of a design method that maximizes the surface areacoverage follows. With reference to FIGS. 6 and 7, the focal length oflens 118 is selected to be equal to one fourth of the diagonal length Dof the active area of detector 84 divided by the tangent of thedivergence angle α of the diffractive optical element 80. Forillustrative clarity, the diagonal length D is shown as segment AB inFIG. 7. Thus: $\begin{matrix}{f_{4} = \frac{D}{4\quad \tan \quad \alpha}} & (6)\end{matrix}$

[0075] The front lens 116 is chosen to produce an overall systemmagnification equivalent to the diagonal length d_(i) of the inputaperture 112 (shown in FIG. 6) divided by the diagonal length D of thedetector array 84. Thus: $\begin{matrix}{f_{3} = {\frac{d_{i}}{D}f_{4}}} & (7)\end{matrix}$

[0076] The overall length L of the imaging portion 54 is given by:$\begin{matrix}{L = {{2\left( {f_{3} + f_{4}} \right)} = \frac{\left( {d_{i} + D} \right)}{2\quad \tan \quad \alpha}}} & (8)\end{matrix}$

[0077] According to an exemplary embodiment of the invention, theaperture(s) 121 may be selected so that the diagonal length d_(i) issubstantially equal to the diagonal length D of the detector array 84(i.e., d_(i)=D). According to such an embodiment, focal length f₃ isequal to focal length f₄ and the overall system length L is given by:$\begin{matrix}{L = {{2\left( {f_{3} + f_{4}} \right)} = \frac{D}{\tan \quad \alpha}}} & (9)\end{matrix}$

[0078] It can be seen from Equations 7 and 8 that in many embodiments itis desirable to have a large diffractive optic divergence angle α toreduce the overall size of imaging portion 54. In practice, divergenceangles α of 5 degrees to 10 degrees produce a relatively compact system.

[0079] In addition to the real-time interferometer embodimentillustrated in FIG. 5, exemplary measurement system 50 of the presentinvention may be configured in a plurality of additional preferredembodiments each designed to carry out a particular type of real-timemeasurement, including a profilometer, a displacement sensor, and awavefront sensor, each of which is described in detail below.

[0080] Referencing FIG. 10, exemplary measurement system 50 of thepresent invention is configured to perform profilometry. Exemplaryprofilometer 50 is configured to perform on-axis illumination andviewing, which is useful in obtaining three-dimensional (3D) informationof the object 60. Many industries utilize profilometry in research anddevelopment, quality control, and manufacturing, including thesemiconductor and medical industries.

[0081] Exemplary transmit portion 52 includes the laser 98 whichtransmits the coherent light wavefront 102. A single polarizingwavefront splitter (PBS) 120 is shared by both the transmit and imageportions 52 and 54 for splitting the light wavefront 102 into thereference wavefront 56 and the object wavefront 58 and combining thereference wavefront 56 and the object wavefront 58 into the combinedwavefront 66. In addition to PBS 120, exemplary image portion 54 of theprofilometer includes the convex lens 116, the diffractive opticalelement 80, the collimating lens 118 displaced from element 80 by itsfocal length, the phase-retardation/-interference plate 96, and the CCDcamera. The computer 62 may be connected to both the transmit and imageportions 52 and 54 to control the operation of the laser 98 and toreceive imaging data 78 from the detector array 84.

[0082]FIG. 11 illustrates an exemplary commercial embodiment of theprofilometer 50 of FIG. 10. As shown, the laser 98 provides the lightwavefront to an integrated measuring unit 122 by means of an opticalcable 124. The integrated measuring unit 122 includes a housing 126 inwhich the common PBS 120, as well as each of the elements of the imageportion 54 shown in FIG. 9, is received. The integrated measuring unit122 transmits and receives the object wavefront 58, with the detectorarray 84 providing image data to the computer 62 via a cable 128.

[0083] Referencing FIG. 12, another exemplary commercial embodiment ofthe measurement system 50 of the present invention is shown andconfigured to function as a displacement sensor. Displacement sensorsare useful in measuring, for example, the vibration or the strain of anobject. Exemplary transmit portion 52 of the displacement-sensorembodiment of the measuring system 50 includes the laser 98 whichtransmits the coherent light wavefront to a fiber wavefront splitter 130via an optical cable 132. The fiber wavefront splitter 130 splits thelight wavefront into the reference wavefront 56, which is provided tothe image portion 52 by an optical cable 134, and the object wavefront58, which is provided to an optics unit 136 by an optical cable 138. Theoptical unit 136 of the transmit portion 52 includes thewavefront-expanding optics of the concave lens 106 and collimating lens108 (see FIG. 5). The operation of the displacement sensor illustratedin FIG. 12 is analogous to that described above.

[0084] According to the displacement-sensor embodiment of themeasurement unit 50, the separate and portable optics unit 136 may bepositioned relative to the test object 60 and the image portion 54. Theobject wavefront 58 can thus be directed to the object 60 from any angleor position.

[0085] Referencing FIG. 13, yet another exemplary commercial embodimentof the measurement system 50 of the present invention is shown andconfigured to function as a wavefront sensor. Wavefront sensors may beused to measure, for example, pressure, temperature, or densitygradients in transparent solids, liquids, or gases. Exemplary transmitportion 52 may include an integrated transmit unit 140 with a housing142, and exemplary image portion 54 may include an integrated receiveunit 144 with a housing 146. Similar to the layout of the measurementsystem 50 shown in FIG. 5, exemplary transmit unit 140 of thewavefront-sensor embodiment of the measuring system 50 includes thelaser which transmits the reference wavefront 56 to the integratedreceive unit 144 via an optical cable 148 and the object wavefront 58 tothe test object 60. The operation of the wavefront sensor illustrated inFIG. 13 is analogous to that described above.

[0086] For each of the foregoing embodiments of the measuring system 50of the present invention, a software application may be utilized by thecomputer 62 for data acquisition and processing. The softwareapplication causes the computer 62 to acquire, process, analyze, anddisplay data associated with the phase-shifted interferograms 74. Dataacquisition may be accomplished by recording two interferograms for eachmeasurement: a reference interferogram for the reference wavefront 56and an object interferogram for the object wavefront 58. Wrapped phasemaps are calculated for each of the interferograms and then subtractedfrom each other. The result is unwrapped to yield a map of the phasechange between the reference and object interferograms. Unwrapping isthe procedure used to remove the modulo 2π ambiguity that ischaracteristic of interferometric data.

[0087] Phase may be calculated based on a single frame of data accordingto:

Φ(x,y)=tan⁻¹ {[I ₃(x,y)−I ₁(x,y)]÷[I ₀(x,y)−I ₂(x,y)],  (10)

[0088] where I₀, I₁, I₂, and I₃ are the respective intensities of eachof the phase-shifted interferograms 74 a-74 d incident on the activesurface 86 of the detector array 84 from the four sections 82 a-82 d(i.e., quadrants Q₀, Q₁, Q₂, and Q₃) as calculated in Equations 4a-4dabove. The variables x and y are the pixel coordinates. To reduce noisein the image, spatial averaging may be used to smooth the phase mapwhile retaining a sharp transition at the 2π−0 phase step. The spatiallyaverages phase may be calculated using the following equations:

Φ(x,y)=tan⁻¹{sum(x,yεδ)[I ₃(x,y)−I ₁(x,y)]÷sum(x,yεδ)[I ₀(x,y)−I₂(x,y)]},  (11)

[0089] where the sums are performed over the range of δ nearestneighbors. Increasing the number of averaged pixels improves smoothnessof the phase map at the expense of spatial resolution; however, thesharpness of the phase discontinuity is retained, thereby permittingrapid phase unwrapping. The unwrapping of phase maps removes thediscontinuous step and permits quantitative analysis of the images.

[0090] The number of pixels averaged may be selected by a user. Forcomparing two states of the system of to subtract background phase noisefrom the system, the phase difference mode can be used. Phase may becalculated according to:

ΔΦ(x,y)=tan⁻¹ [X(x,y)÷Y(x,y)],  (12)

[0091] where:

X(x,y)=[Ib ₃(x,y)−Ib ₁(x,y)]*[It ₀(x,y)−It ₂(x,y)]−[It ₃(x,y)−It₁(x,y)]*[Ib ₀(x,y)−Ib ₂(x,y)],

Y(x,y)=[Ib ₀(x,y)−Ib ₂(x,y)]*[It ₀(x,y)−It ₂(x,y)]+[Ib ₃(x,y)−Ib₁(x,y)]*[It ₃(x,y)−It ₁(x,y)],

[0092] Ib is the baseline image captured, and

[0093] It is the image captured for comparison.

[0094] Spatial averaging can be accomplished using the formula:

ΔΦ(x,y)=tan⁻¹[sum(x,yεδ)X(x,y)÷sum(x,yεδ)Y(x,y)].  (13)

[0095] The three dimensional shape of an object can be determined byusing two color interferometry. To do so, a first set of fourphase-shifted interferograms is captured at a first wavelength λ₁ (i.e.,Ib_(n)), and a second set of phase-shifted interferograms is captured ata second wavelength λ₂ (i.e., It_(n)). The relative distance to theobject (or range) is calculated by: $\begin{matrix}{{{R\left( {x,y} \right)} = {\frac{\lambda^{2}}{4{\pi\Delta\lambda}}{\tan^{- 1}\left( \frac{X\left( {x,y} \right)}{Y\left( {x,y} \right)} \right)}}},} & (14)\end{matrix}$

[0096] where:

X(x,y)=[Ib ₃(x,y)−Ib ₁(x,y)]*[It ₀(x,y)−It ₂(x,y)]−[It ₃(x,y)−It₁(x,y)]*[Ib ₀(x,y)−Ib ₂(x,y)]

Y(x,y)=[Ib ₀(x,y)−Ib ₂(x,y)]*[It ₀(x,y)−It ₂(x,y)]+[Ib ₃(x,y)−Ib₁(x,y)]*[It ₃(x,y)−It ₁(x,y)]

[0097] Noise in the image can be significantly reduced using a weightedspatial average over neighboring pixels. This can be accomplished by:$\begin{matrix}{{{R\left( {x,y} \right)} = {\frac{\lambda^{2}}{4{\pi\Delta\lambda}}{\tan^{- 1}\left( \frac{\sum\limits_{x,{y \in \delta}}{X\left( {x,y} \right)}}{\sum\limits_{x,{y \in \delta}}{Y\left( {x,y} \right)}} \right)}}},} & (15)\end{matrix}$

[0098] where the sums are performed over the range of δ nearestneighbors. Because of the modelo 2πbehavior of the arc tangent function,the range is wrapped (ambiguous) beyond the so-called syntheticwavelength of: $\begin{matrix}{\lambda_{s} = {\frac{\lambda^{2}}{4{\pi\Delta\lambda}}.}} & (16)\end{matrix}$

[0099] The well-known process of spatial phase unwrapping can be used toremove the discontinuous steps and to permit quantitative analysis ofthe images. Alternatively, it is possible to use multiple syntheticwavelengths and incrementally add the range distance as known in theart. The overall range is then given by: $\begin{matrix}{{{R^{\prime}\left( {x,y} \right)} = {\sum\limits_{m}\frac{R_{\Delta \quad {\lambda m}}\left( {x,y} \right)}{m}}},} & (17)\end{matrix}$

[0100] where m is the number of wavelength steps used and R_(Δλm) is therange measured with a frequency tuning of Δλ/m. Implied in this methodis that no single measurement should have a phase value greater than 2π,which can place a restriction on the maximum size of the object that canbe measured.

[0101] Referencing FIG. 14, a user interface 148 provided by thesoftware of the invention is shown displaying a raw interferogram 150and wrapped phasemaps 152 from a central portion of the rawinterferogram 150. The raw interferogram 150 illustrates data 78resulting from the measurement of a diffusion flame.

[0102] Those skilled in the art will understand that the precedingexemplary embodiments of the present invention provide the foundationfor numerous alternatives and modifications thereto. These othermodifications are also within the scope of the present invention.Accordingly, the present invention is not limited to that precisely asshown and described above.

What is claimed is:
 1. Apparatus for splitting a wavefront, saidapparatus comprising: a wavefront-splitting element for: receiving awavefront; splitting said wavefront into a plurality of sub-wavefronts;imaging said sub-wavefronts such that of each of said imagedsub-wavefront is substantially contiguous with at least one other saidimaged sub-wavefront; providing said imaged sub-wavefronts; a sensingelement for receiving said imaged sub-wavefronts from saidwavefront-splitting element.
 2. Apparatus as claimed in claim 1 whereinsaid sensing element is a detector array.
 3. Apparatus as claimed inclaim 1 wherein said sensing element has a surface area, said pluralityof sub-wavefronts being incident on a substantial portion of saidsurface area of said sensing element.
 4. Apparatus as claimed in claim 1wherein said wavefront-splitting element splits said wavefront into foursub-wavefronts.
 5. Apparatus as claimed in claim 1 wherein saidwavefront-splitting element includes a diffractive optical element. 6.Apparatus as claimed in claim 1 wherein said wavefront-splitting elementand said sensing element are positioned with respect to each other sothat an optical axis is defined between said elements and issubstantially normal to said elements.
 7. Apparatus as claimed in claim6 wherein said wavefront-splitting element and said sensing element arepositioned along said optical axis such that said sub-wavefronts areincident on said sensing element at substantially the same time. 8.Apparatus as claimed in claim 6 wherein said wavefront-splitting elementsplits said wavefront such that said sub-wavefronts diverge from saidoptical axis at substantially the same angle.
 9. Apparatus as claim 6wherein each of said sub-wavefronts has an optical path defined betweensaid wavefront-splitting element and said sensing element; said opticalpaths of said sub-wavefronts having substantially the same length. 10.Apparatus as claimed in claim 6 further comprising a collimating lenspositioned between said wavefront-splitting element and said sensingelement; said collimating lens for collimating said sub-wavefronts. 11.Apparatus as claimed in claim 10 wherein said collimating lens has afocal length; said wavefront-splitting element and said sensing elementare positioned from said collimating lens by a distance substantiallyequal to said focal length.
 12. Apparatus as claimed in claim 10 furthercomprising an input lens having a focal length and being positioned fromsaid wavefront-splitting element by a distance substantially equal tosaid focal length.
 13. Apparatus as claimed in claim 12 furthercomprising an aperture positioned from said input lens by a distancesubstantially equal to said focal length.
 14. Apparatus as claimed inclaim 1 wherein said wavefront includes a reference wavefront and anobject wavefront, said reference wavefront and said object wavefrontbeing orthogonally polarized; said wavefront-splitting element splittingsaid wavefront such that each of said sub-wavefront includes saidreference wavefront and said object wavefront.
 15. Apparatus as claimedin claim 14 further comprising a phase-shifting interference elementpositioned between said wavefront-splitting element and said sensingelement, said phase-shifting interference element for: shifting therelative phase between said reference wavefront and said objectwavefront of said sub-wavefronts to yield a respective plurality ofphase-shifted sub-wavefronts; interfering said reference and said objectwavefronts of said phase-shifted sub-wavefronts to yield a respectiveplurality of phase-shifted interferograms; said phase-shiftedinterferograms being incident on said sensing element.
 16. A method forsplitting a wavefront, said method comprising the steps of: receiving awavefront; splitting said wavefront into a plurality of sub-wavefronts;and imaging said sub-wavefronts such that each of said imagedsub-wavefronts is substantially contiguous with at least one other saidimaged sub-wavefront.
 17. A method as claimed in claim 16 wherein saidsplitting step comprises the step of: splitting said wavefronts suchthat said sub-wavefronts are incident on a sensing element substantiallysimultaneously.
 18. A method as claimed in claim 16 further comprisingthe step of: collimating said sub-wavefronts.
 19. A method as claimed inclaim 18 further comprising the step of: sensing said sub-wavefrontsafter said collimating step.
 20. A method as claimed in claim 19 whereinsaid collimating step comprises the step of: collimating saidsub-wavefronts such that said sub-wavefronts are incident on asubstantial portion of a sensing element.
 21. A method as claimed inclaim 20 wherein said splitting step comprises the step of: splittingsaid wavefront such that said sub-wavefronts respectively have opticalpaths to said sensing element of substantially the same length.
 22. Amethod as claimed in claim 16 wherein splitting step comprises the stepof: splitting said wavefront into four sub-wavefronts.
 23. A method asclaimed in claim 22 further comprising the step of: sensing saidsub-wavefronts with a single detector array.
 24. A method as claimed inclaim 23 wherein said splitting step comprises the step of: splittingsaid wavefront such that said sub-wavefronts are imaged with a singledetector array.
 25. A method as claimed in claim 16 wherein saidsplitting step comprises the step of: splitting said wavefront such thatsub-wavefronts diverge from an optical axis of said wavefront atsubstantially the same angle.
 26. A method as claimed in claim 16further comprising the step of transmitting a first wavefront at a firstwavelength to an object, wherein: said receiving step comprises the stepof receiving said first wavefront from said object; and said splittingstep comprises the step of splitting said first wavefront into a firstset of sub-wavefronts such that each is substantially contiguous with atleast one other said sub-wavefront; further comprising the step ofsensing said first set of sub-wavefronts.
 27. A method as claimed inclaim 26 further comprising the step of: transmitting a second wavefrontat a second wavelength to the object; receiving said second wavefrontfrom the object; splitting said second wavefront into a second set ofsub-wavefronts such that each is substantially contiguous with at leastone other said sub-wavefront; and sensing said second set ofsub-wavefronts.
 28. A method as claimed in claim 27 further comprisingthe steps of: determining the distance to the object based on said firstand second sets of sub-wavefronts.